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Matching van Stockum dust to Papapetrou vacuum

Michal Marvan

Abstract

Addressing a long-standing problem, we show that every van Stockum dust can be matched to a 1-parametric family of non-static Papapetrou vacuum metrics, and the converse. The boundary, if existing, is determined by vanishing of certain first-order invariant on the vacuum side. Moreover, we establish a relation to Ehlers and Kramer--Neugebauer transformations, which allows us to look for dust clouds with a prescribed boundary. Explicit examples include the Bonnor metric and a new vacuum exterior to the Lanczos--van Stockum dust metric, as well as dust clouds with nontrivial topology.

Matching van Stockum dust to Papapetrou vacuum

Abstract

Addressing a long-standing problem, we show that every van Stockum dust can be matched to a 1-parametric family of non-static Papapetrou vacuum metrics, and the converse. The boundary, if existing, is determined by vanishing of certain first-order invariant on the vacuum side. Moreover, we establish a relation to Ehlers and Kramer--Neugebauer transformations, which allows us to look for dust clouds with a prescribed boundary. Explicit examples include the Bonnor metric and a new vacuum exterior to the Lanczos--van Stockum dust metric, as well as dust clouds with nontrivial topology.
Paper Structure (12 sections, 8 theorems, 36 equations, 1 figure, 1 table)

This paper contains 12 sections, 8 theorems, 36 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Matching
  3. The choice of coordinates
  4. A useful invariant
  5. Van Stockum metrics
  6. Papapetrou metrics
  7. Locating the boundary
  8. Electrostatic analogy
  9. Relation to static Weyl vacuum metrics
  10. Examples
  11. Discussion
  12. Visualisation

Key Result

Proposition 1

Assuming $f_1^{\rm (I)} \equiv_{B}^{k} f_1^{\rm (II)}$, $\dots$, $f_m^{\rm (I)} \equiv_{B}^{k} f_m^{\rm (II)}$, let $F(f_1,\dots,f_m)$ be a $C^k$-continuous function in a neighbourhood of the image $f_1^{\rm (I)} B \times \cdots \times f_m^{\rm (I)} B = f_1^{\rm (II)} B \times \cdots \times f_m^{\ holds.

Figures (1)

  • Figure 1: Axisymmetric dust clouds examples in Weyl's coordinates. (1) Densities (increasing from dark to light). (2) Admissible boundaries. (3) Example clouds.

Theorems & Definitions (15)

  • Proposition 1
  • Proposition 2
  • proof
  • Corollary 1
  • Definition 1
  • Theorem 1
  • proof
  • Proposition 3
  • proof
  • Corollary 2
  • ...and 5 more